Deriving the Area of a Circle — Another Way

In the previous post, we have learned how to find the area of a circle by dividing it into congruent sectors. We have seen that we can divide the circle into as many sectors as we can to better approximate the area of a circle.  In this post, we find the area of the circle by making concentric circular cuts forming thin strips and “bending” these strips straight.

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If we make the cut as thin as possible, we can form a triangle by ordering these strips by length. For instance, the circumference of the outermost circle is the longest and as we cut the strips of the inner circles, the circumference gradually becomes shorter. If the cut strips indeed form a triangle, then, the base of the triangle is equal to the circumference of the circle with radius r which equals 2 \pi r.  Its height is r.

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Since the area of a triangle is the half the product of the base and its height, the area A of the triangle above, which is also the area of the circle, is

A = \displaystyle\frac{1}{2}(2 \pi r)(r) = \pi r^2.

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